ORELA Mathematics (NT304) flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Discriminant

For ax² + bx + c = 0, the value b² − 4ac. Positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex conjugate roots.

Quadratic formula

The solutions of ax² + bx + c = 0 with a ≠ 0 are x = [−b ± √(b² − 4ac)]/(2a). It solves every quadratic, including those that do not factor over the integers.

Vertex of a parabola

For y = ax² + bx + c, the vertex is at x = −b/(2a). The parabola opens upward when a > 0 and downward when a < 0, and the vertex is the minimum or maximum.

Slope formula

The slope through (x₁, y₁) and (x₂, y₂) is (y₂ − y₁)/(x₂ − x₁), the vertical change divided by the horizontal change. A vertical line has undefined slope; a horizontal line has slope zero.

Perpendicular slope rule

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so m₁·m₂ = −1. A horizontal line is perpendicular to a vertical line, whose slope is undefined.

Point-slope form

A line with slope m through (x₁, y₁) satisfies y − y₁ = m(x − x₁). Substituting a known point and slope gives the equation directly without first finding the intercept.

Distance and midpoint formulas

For points (x₁, y₁) and (x₂, y₂), distance is √[(x₂ − x₁)² + (y₂ − y₁)²] and midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). Both follow from the Pythagorean theorem.

Rational exponent rule

For positive a, a^(m/n) equals the nth root of a raised to the m, that is ⁿ√(aᵐ). The denominator of the exponent is the index of the radical and the numerator is the power.

Negative exponent rule

For any nonzero base a, a⁻ⁿ = 1/aⁿ, so a negative exponent indicates a reciprocal rather than a negative result. For example, 2⁻³ = 1/8, not −8.

Logarithm product and quotient laws

For positive arguments, log(xy) = log x + log y and log(x/y) = log x − log y. These laws let you expand or condense logarithmic expressions with the same base.

Change of base formula

log_b x = (log_c x)/(log_c b) for any valid base c. It converts a logarithm in an unavailable base into a quotient of common or natural logarithms for computation.

Inverse function condition

A function has an inverse only if it is one-to-one, meaning distinct inputs give distinct outputs. The horizontal line test detects this, and the inverse graph reflects the original across y = x.

Composition of functions

The composition (f ∘ g)(x) equals f(g(x)), applying g first and then f. Composition is generally not commutative, so f(g(x)) and g(f(x)) usually differ.

Arithmetic sequence formulas

With first term a₁ and common difference d, the nth term is a₁ + (n − 1)d and the sum of n terms is n(a₁ + aₙ)/2. Each term adds the same constant.

Geometric series sum

A geometric series with first term a₁ and ratio r sums to a₁(1 − rⁿ)/(1 − r) for n terms when r ≠ 1. If |r| < 1, the infinite sum is a₁/(1 − r).

Fundamental counting principle

If one choice can be made in m ways and a second in n ways, the two together can be made in m·n ways. This multiplication rule extends to any number of independent stages.

Permutation formula

The number of ordered arrangements of r objects chosen from n distinct objects is nPr = n!/(n − r)!. Order matters, so ABC and CBA count as different arrangements.

Combination formula

The number of unordered selections of r objects from n distinct objects is nCr = n!/[r!(n − r)!]. Order does not matter, so ABC and CBA count as the same selection.

Probability of a complement

For any event E, P(not E) = 1 − P(E). The complement rule is often the fastest route when counting the opposite outcome is simpler than counting the event itself.

Addition rule for probability

P(A or B) = P(A) + P(B) − P(A and B). Subtract the overlap so outcomes in both events are not counted twice. For mutually exclusive events the overlap is zero.

Conditional probability

P(A|B) = P(A and B)/P(B), provided P(B) > 0. It gives the probability of A restricted to the reduced sample space in which B has already occurred.

Independent events

Events A and B are independent exactly when P(A and B) = P(A)·P(B), equivalently P(A|B) = P(A). Knowing one occurred does not change the other's probability.

Mean, median, and mode

The mean is the sum divided by the count, the median is the middle value of ordered data, and the mode is the most frequent value. The median resists outliers more than the mean.

Standard deviation

The square root of the variance, measuring typical distance of data values from the mean. A larger standard deviation means the data are more spread out around the mean.

Normal distribution empirical rule

About 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. The distribution is symmetric about its mean.

Triangle angle sum

The interior angles of any Euclidean triangle add to 180°, and an exterior angle equals the sum of the two nonadjacent interior angles. In a polygon with n sides the interior sum is (n − 2)·180°.

Pythagorean theorem

In a right triangle with legs a and b and hypotenuse c, a² + b² = c². The converse holds: if the sides satisfy this equation, the triangle is right.

Area and circumference of a circle

A circle of radius r has area πr² and circumference 2πr. A sector with central angle θ in degrees has area (θ/360)·πr² and arc length (θ/360)·2πr.

Volume formulas

A cylinder has volume πr²h, a cone (1/3)πr²h, a sphere (4/3)πr³, and a rectangular prism lwh. Cones and pyramids carry a one-third factor relative to matching prisms.

Similar figures ratio rule

If two similar figures have linear scale factor k, their areas scale by k² and their volumes by k³. Corresponding angles are equal and corresponding side lengths are proportional.

Derivative power rule

For f(x) = xⁿ, the derivative is f′(x) = n·xⁿ⁻¹. The derivative gives the slope of the tangent line and the instantaneous rate of change of the function.

Definite integral as area

The definite integral of a nonnegative function from a to b equals the area under its curve above the x-axis. The Fundamental Theorem evaluates it as F(b) − F(a) for an antiderivative F.